President Vladimir V. Putin of Russia on Friday insisted that Ukraine order some of its forces to surrender to Russia, a striking demand made hours after President Trump said the United States had “very good and productive” discussions with Mr. Putin about a potential cease-fire in Ukraine.

Mr. Putin’s televised comments came shortly after Mr. Trump, on social media, said he had urged the Russian leader to spare the lives of Ukrainian soldiers struggling to hold on to a patch of land in the Kursk region of Russia.

“I have strongly requested to President Putin that their lives be spared,” Mr. Trump wrote.

Both presidents claimed on Friday that Ukrainian forces were surrounded in Kursk, the area where Kyiv’s troops stunned Russia with a cross-border incursion last summer. Independent analysts have challenged those claims, and Ukraine’s military on Friday again rejected them.

Still, Russian forces have of late had the upper hand in the fighting in Kursk. And Mr. Putin said that for Mr. Trump’s call “to be effectively implemented,” the leaders of Ukraine needed to order “its military units to lay down their arms and surrender.” Neither man has raised the idea of Russian troops on Ukrainian land surrendering.

Battles have been raging in Kursk as Moscow’s forces push to drive Ukrainian troops from the land they seized over the summer. Russian troops have advanced in recent days, with Mr. Putin urging them to finish the job “in the shortest possible time.”

What happens next in Kursk has become a key point of contention ahead of a potential cease-fire. Mr. Putin had on Thursday suggested less explicitly that he wanted Ukraine to order its soldiers there to surrender as part of any deal to end the war, which has killed more than a million soldiers and countless civilians over three years of grinding fighting.

But his comments on Friday were far more direct, and came as both the Kremlin and White House signaled negotiations were moving forward despite the raft of concessions Moscow appears to be seeking.

After the Russian leader met with Steve Witkoff, the U.S. special envoy, in Moscow, late Thursday, top officials at both the White House and the Kremlin expressed “cautious optimism.”

“We had very good and productive discussions with President Vladimir Putin of Russia yesterday,” Mr. Trump wrote Friday morning on his Truth Social platform, in an apparent reference to the meeting with Mr. Witkoff. “There is a very good chance that this horrible, bloody war can finally come to an end.”

But President Volodymyr Zelensky of Ukraine questioned whether Russia was acting in good faith, pointing out that his country had already agreed to a 30-day truce after a meeting with U.S. officials in Saudi Arabia this week.

“The world sees how Russia is deliberately setting conditions that only complicate and drag out the process,” Mr. Zelensky said Friday on social media. “Russia is the only party that wants the war to continue and diplomacy to break down.”

It wasn’t immediately clear whether Mr. Putin and Mr. Trump had directly spoken to each other. The Kremlin said earlier Friday that Mr. Putin, in his meeting with Mr. Witkoff, “passed along information and additional signals for President Trump.” It also said that Mr. Putin expected to talk to Mr. Trump but that the call had yet to be scheduled.

In that Truth Social post, Mr. Trump also said thousands of Ukrainian troops were “completely surrounded by the Russian military” — an apparent reference to Russian claims that Ukrainian soldiers were surrounded in Kursk

“There is no threat of encirclement of our units,” the Ukrainian military’s general staff said in a statement soon after on Friday, calling such reports “false and fabricated by the Russians.”

A Ukrainian soldier fighting in Kursk who was reached by phone on Friday, speaking on condition of anonymity to discuss the current situation on the battlefield, said “the situation is bad, almost critical.” But it was not as bad as Mr. Trump described in his post, the soldier added.

The Ukrainian authorities on Friday ordered several villages in the Sumy region, across the border from Kursk, to evacuate amid increased attacks and fears that the fighting could spill over.

“Aerial attacks — such as glide bombs and drones — have intensified in the border areas,” Volodymyr Artiukhin, the Ukrainian head of the Sumy Regional Military Administration, said on Facebook. He announced the mandatory evacuation on Friday of eight villages, affecting 543 residents.

Driving Ukrainian troops out of Kursk would deny Mr. Zelensky a significant bargaining chip in any negotiations. And before the meeting with Mr. Witkoff, Mr. Putin showed on Thursday that he was in no hurry to accept the offer of a 30-day truce made by Ukraine and the United States this week — telling a news conference that he was open to the proposal, but adding that he would seek to negotiate over a slew of issues, such as Western weapons deliveries to Ukraine, that could delay or derail any deal.

Dmitri S. Peskov, Mr. Putin’s spokesman, suggested on Friday that the outcome of the diplomatic back-and-forth would only become clear after Mr. Witkoff had briefed Mr. Trump and after the Russian and American leaders had spoken by phone. The two leaders are last known to have spoken on Feb. 12.

Mr. Witkoff, who is officially the president’s Middle East envoy, has also taken on a key role as an interlocutor with Russia — spending three hours meeting with Mr. Putin last month as he finalized a prisoner exchange.

“After Mr. Witkoff passes along all of the information he received in Moscow to his head of state — we’ll determine the timing of the conversation after that,” Mr. Peskov said. “There’s an understanding on both sides that such a conversation is necessary.”

Mr. Peskov’s comments were the latest indication that Mr. Putin is likely trying to balance a desire to avoid upsetting Mr. Trump with his effort to force wide-ranging concessions from the West and from Ukraine. While Mr. Trump says he wants to end the war as quickly as possible, Mr. Putin appears confident that he has time on his side and that an unconditional cease-fire would benefit Ukraine.

Officials in Kyiv have argued that Mr. Putin is now simply trying to drag out negotiations while he continues fighting and has no plans to make concessions that could bring about a durable peace.

“Ukraine seeks an end to the war,” the country’s foreign minister, Andrii Sybiha, posted on X on Friday. “Putin seeks to continue the war. The rest of his words are just a smoke-screen.”

Marc Santora and contributed reporting from Dnipro, Ukraine.